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**Supporting Rigorous Mathematics Teaching and Learning**

**Selecting and Sequencing Based on Essential Understandings**

Tennessee Department of Education Middle School Mathematics Grade 8 © 2013 UNIVERSITY OF PITTSBURGH

**Rationale**

There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). By engaging in an analysis of a lesson-planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding.

2 © 2013 UNIVERSITY OF PITTSBURGH

**Session Goals**

Participants will learn about: • goal-setting and the relationship of goals to the CCSS and essential understandings; • essential understandings as they relate to selecting and sequencing student work; • Accountable Talk ® moves related to essential understandings; and • prompts that problematize or “hook” students during the Share, Discuss, and Analyze phase of the lesson.

Accountable Talk ® is a registered trademark of the University of Pittsburgh.

© 2013 UNIVERSITY OF PITTSBURGH 3

### “The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.”

Brahier, 2000 4

“During the planning phase, teachers make decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals.” Fennema & Franke, 1992, p. 156 5

**Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework**

**TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning**

Stein, Smith, Henningsen, & Silver, 2000 6

**Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework**

**TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students**

Setting Goals Selecting Tasks Anticipating Student Responses

**Student Learning**

Stein, Smith, Henningsen, & Silver, 2000 Orchestrating Productive Discussion • Monitoring students as they work • Asking assessing and advancing questions • Selecting solution paths • Sequencing student responses • Connecting student responses via

*Accountable Talk*

discussions 7

**Identify Goals for Instruction and Select an Appropriate Task**

© 2013 UNIVERSITY OF PITTSBURGH 8

**The Structure and Routines of a Lesson**

**The Explore Phase/Private Work Time**

Generate Solutions

**The Explore Phase/ Small Group Problem Solving**

1. Generate and Compare Solutions 2. Assess and Advance Student Learning

**Share, Discuss, and Analyze Phase of the Lesson**

1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write © 2013 UNIVERSITY OF PITTSBURGH

**MONITOR**

: Teacher selects examples for the Share, Discuss, • • • and Analyze Phase based on: • Different solution paths to the same task Different representations Errors Misconceptions

**SHARE**

: Students explain their methods, repeat others ’ ideas, put ideas into their own words, add on to ideas and ask for clarification.

*REPEAT THE CYCLE FOR EACH SOLUTION PATH*

**COMPARE**

: Students discuss similarities and difference between solution paths.

**FOCUS: **

Discuss the meaning of mathematical ideas in each representation

**REFLECT**

: Engage students in a Quick Write or a discussion of the process.

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**Contextualizing Our Work Together**

Imagine that you are working with a group of students who have the following understanding of the concepts. • 70% of the students need to make sense of what it means to solve a system of equations. (8.EE.C.8, C.8a, C.8b) • 20% of the students need additional work on using linear equations to model a problem situation. (8.F.B.4) • Five percent of the students still struggle with solving linear equations in one variable. (8.EE.C.7) • Five percent of the students struggle to pay attention and their understanding of expressions and equations is two grade levels below eighth grade.

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**The CCSS for Mathematics: Grade 8**

**Expressions and Equations 8.EE**

**Analyze and solve linear equations and pairs of simultaneous linear equations.**

8.EE.C.7

Solve linear equations in one variable.

8.EE.C.8

Analyze and solve pairs of simultaneous linear equations.

8.EE.C.8a

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8b

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

*For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.*

8.EE.C.8c

Solve real-world and mathematical problems leading to two linear equations in two variables.

*For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.*

11 Common Core State Standards, 2010, p. 54 - 55, NGA Center/CCSSO

**The CCSS for Mathematics: Grade 8**

**Functions Use functions to model relationships between quantities.**

8.F.B.4

**8.F**

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (

*x, y*

) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

12 Common Core State Standards, 2010, p. 55, NGA Center/CCSSO

**Standards for Mathematical Practice Related to the Task**

**1.**

**2.**

**3.**

**4.**

5.

**6.**

**7.**

8.

**Make sense of problems and persevere in solving them.**

**Reason abstractly and quantitatively.**

**Construct viable arguments and critique the reasoning of others.**

**Model with mathematics.**

Use appropriate tools strategically.

**Attend to precision.**

**Look for and make use of structure.**

Look for and express regularity in repeated reasoning.

Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 13

**Identify Goals: Solving the Task**

**(Small Group Discussion)**

Solve the task.

Discuss the possible solution paths to the task. © 2013 UNIVERSITY OF PITTSBURGH 14

**Scuba Math Task**

Serena and Trevon are taking a scuba diving course while on vacation in Hawaii. Serena begins swimming toward the surface as Trevon begins his dive. The tables below represent their depth in feet with respect to time in seconds.

**Time (seconds)**

0 10 20 30 40

**Serena’s Depth (feet)**

-90 -85 -80 -75 -70

**Time (seconds)**

0 20 40 60 80

**Trevon’s Depth (feet)**

0 -25 -50 -75 -100 At what time will Trevon and Serena be at the same depth? Show your work and explain your reasoning. © 2013 UNIVERSITY OF PITTSBURGH 15

**Identify Goals Related to the Task**

**(Whole Group Discussion)**

Does the task provide opportunities for students to access the Standards for Mathematical Content and the Standards for Mathematical Practice that we have identified for student learning?

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**Identify Goals: Essential Understandings **

**(Whole Group Discussion)**

Study the essential understandings associated with the Expressions and Equations Common Core Standards. Which of the essential understandings are the goals of the Scuba Math Task?

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**Essential Understandings **

**(Small Group Discussion) Essential Understanding Solutions Make the Equations True**

The solution(s) of a system of two linear equations is the ordered pair or pairs (

*x, y*

) that satisfy both equations.

**Systems of Equations Can be Solved Graphically**

The line representing a linear equation consists of all of the ordered pairs (

*x, y*

) that satisfy the equation. So, the solution of a system of linear equations is represented graphically by the intersection of the lines representing the equations because the point(s) at the intersection satisfy both equations.

**Systems of Equations Can be Solved Algebraically Using Substitution**

Given a true equation in two variables, the equation formed by isolating either of the variables is also true. Therefore, so is the equation you get by substituting an expression equal to one of the variables into a second true equation.

**A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions**

Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution. Parallel lines have no points in common. Therefore, a system of two linear equations representing distinct parallel lines has no solutions.

Linear equations representing the same line have infinitely many points in common.

Therefore, a system of two linear equations representing the same line has infinitely many solutions.

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**Selecting and Sequencing Student Work for the Share, Discuss, and Analyze Phase of the Lesson**

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**Analyzing Student Work**

**(Private Think Time)**

• Analyze the student work on pages 12-17 of the your handout. • Identify what each group knows related to the essential understandings.

• Consider the questions that you have about each group’s work as it relates to the essential understandings.

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**Prepare for the Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion)**

Assume that you have circulated and asked students assessing and advancing questions.

Study the student work samples.

1. Which pieces of student work will allow you to address the essential understanding? 2.

How will you sequence the student’s work that you have selected? Be prepared to share your rationale. 21 © 2013 UNIVERSITY OF PITTSBURGH

**The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work**

**(Small Group Discussion)**

In your small group, come to a consensus on the work that you select, and share your rationale. Be prepared to justify your selection and sequence of student work.

**Group(s) Order Rationale Essential Understandings Solutions Make the Equations True Systems of Equations Can be Solved Graphically Systems of Equations Can be Solved Algebraically Using Substitution A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions**

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**The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work**

**(Whole Group Discussion)**

What order did you identify for the EUs and student work? What is your rationale for each selection?

**Essential Understandings #1 via Gr.**

**#2 via Gr.**

**#3 via Gr.**

**#4 Via Gr.**

**Solutions Make the Equations True**

The solution(s)…

**Systems of Equations Can be Solved Graphically**

The line representing…

**Systems of Equations Can be Solved Algebraically Using Substitution**

Given two true equations…

**A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions**

Two distinct lines… 23 © 2013 UNIVERSITY OF PITTSBURGH

**Group A**

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**Group B**

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**Group C**

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**Group D **

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**Group E**

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**Group F**

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**The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work**

**(Whole Group Discussion)**

What order did you identify for the EUs and student work? What is your rationale for each selection?

**Essential Understandings #1 via Gr.**

**#2 via Gr.**

**#3 via Gr.**

**#4 Via Gr.**

**Solutions Make the Equations True**

The solution(s)…

**Systems of Equations Can be Solved Graphically**

The line representing…

**Systems of Equations Can be Solved Algebraically Using Substitution**

Given two true equations…

**A System of Two Linear Equations Can Have Zero, One, or Infinitely Many Solutions**

Two distinct lines… © 2013 UNIVERSITY OF PITTSBURGH 30

**Academic Rigor in a Thinking Curriculum The Share, Discuss, and Analyze Phase of the Lesson**

© 2013 UNIVERSITY OF PITTSBURGH 31

**Academic Rigor In a Thinking Curriculum**

A teacher must always be assessing and advancing student learning. A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson.

*Accountable Talk *

discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson. © 2013 UNIVERSITY OF PITTSBURGH 32

*Accountable Talk *Discussions

*Accountable Talk*Discussions

Recall what you know about the

*Accountable Talk *

features and indicators. In order to recall what you know: • Study the chart with the

*Accountable Talk *

moves. You are already familiar with the

*Accountable Talk*

moves that can be used to

**Ensure Purposeful, Coherent, and Productive Group Discussion.**

• Study the

*Accountable Talk *

moves associated with creating accountability to: the learning community; knowledge; and rigorous thinking.

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*Accountable Talk *Features and Indicators

*Accountable Talk*Features and Indicators

**Accountability to the Learning Community**

• • • • Active participation in classroom talk.

Listen attentively.

Elaborate and build on each others’ ideas.

Work to clarify or expand a proposition.

**Accountability to Knowledge**

• • • Specific and accurate knowledge.

Appropriate evidence for claims and arguments.

Commitment to getting it right.

**Accountability to Rigorous Thinking**

• • • • • Synthesize several sources of information.

Construct explanations and test understanding of concepts.

Formulate conjectures and hypotheses.

Employ generally accepted standards of reasoning.

Challenge the quality of evidence and reasoning.

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*Accountable Talk *Moves

*Accountable Talk*Moves

**Talk Move**

Marking Challenging Revoicing Recapping

**Function To Ensure Purposeful, Coherent, and Productive Group Discussion**

Direct

**attention**

to the value and importance of a student’s contribution.

**Example**

That’s an important point. One factor tells use the number of groups and the other factor tells us how many items in the group.

Let me challenge you: Is that always true?

Redirect a question back to the students or use students’ contributions as a source for further challenge or query.

Align a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content. Make public in a concise, coherent form, the group’s achievement at creating a shared understanding of the phenomenon under discussion.

S: 4 + 4 + 4.

You said three groups of four. Let me put these ideas all together.

What have we discovered?

Keeping the Channels Open Keeping Everyone Together Linking Contributions Verifying and Clarifying

**To Support Accountability to Community**

Ensure that students can hear each other, and remind them that they must hear what others have said.

Ensure that everyone not only heard, but also understood, what a speaker said.

Make explicit the relationship between a new contribution and what has gone before.

Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation.

SAY that again and louder.

Can someone repeat what was just said?

Can someone add on to what was said?

Did everyone hear that?

Does anyone have a similar idea?

Do you agree or disagree with what was said?

Your idea sounds similar to his idea. So are you saying..?

Can you say more? Who understood what was said?

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*Accountable Talk *Moves

*Accountable Talk*Moves

*(continued)*

Pressing for Accuracy Building on Prior Knowledge

**To Support Accountability to Knowledge**

Hold students accountable for the accuracy, credibility, and clarity of their contributions.

Tie a current contribution back to knowledge accumulated by the class at a previous time.

Pressing for Reasoning

**To Support Accountability to Rigorous Thinking**

Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise.

Expanding Reasoning Open up extra time and space in the conversation for student reasoning.

Why does that happen?

Someone give me the term for that.

What have we learned in the past that links with this?

SAY why this works.

What does this mean?

Who can make a claim and then tell us what their claim means?

Does the idea work if I change the context? Use bigger numbers?

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**The Share, Discuss, and Analyze Phase of the Lesson: Planning a Discussion **

**(Small Group Discussion)**

• From the list of potential EUs and its related student work, each group will select an essential understanding to focus their discussion. • Identify a teacher in the group who will be in charge of leading a discussion with the group after the

*Accountable Talk *

moves related to the EU have been written.

*Write a set of *

Accountable Talk

*moves on chart paper so it is public to your group for the next stage in the process. *

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**An Example: ***Accountable Talk *Discussion

*Accountable Talk*Discussion

The Focus Essential Understanding

**A System of Equations May Have Zero, One, or Infinitely Many Solutions**

Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution.

**Group C Group F**

• How did your group determine that Trevon and Serena are moving at different • • • paces? Who understood what she said about Trevon’s and Serena’s paces? (Community) Can you say back what he said about the rate of change? (Community) Who can add on and talk about where we see the pace in the equation and graph? (Community) • • The slope is visible in the table, the equation, and the graph. (Marking) If the pace or slope is different for each person, what does that mean about whether • or not Trevon and Serena are ever at the same depth? (Rigor) What will be true about the slope and the equations if are the same line (second piece of work)? (Rigor) © 2013 UNIVERSITY OF PITTSBURGH Trevon’s and Serena’s data 38

**Problematize the ***Accountable Talk *Discussion

*Accountable Talk*Discussion

**(Whole Group Discussion)**

Using the list of essential understandings identified earlier, write

*Accountable Talk *

discussion questions to elicit from students a discussion of the mathematics.

Begin the discussion with a “hook” to get student attention focused on an aspect of the mathematics.

**Type of Hook**

Compare and Contrast

**Example of a Hook**

Compare the half that has two equal pieces with the figure that has three pieces.

Insert a Claim and Ask if it is True Three equal pieces of the six that are on one side of the figure show half of the figure. If I move the three pieces to different places in the whole, is half of the figure still shaded? Challenge You said two pieces are needed to create halves. How can this be half; it has three pieces?

A Counter-Example If this figure shows halves (a figure showing three sixths), tell me about this figure (a figure showing three sixths but the sixths are not equal pieces).

39 © 2013 UNIVERSITY OF PITTSBURGH

**An Example: ***Accountable Talk *Discussion

*Accountable Talk*Discussion

The Focus Essential Understanding

**A System of Equations May Have Zero, One, or Infinitely Many Solutions**

Two distinct lines will intersect at one point if and only if they do not have the same slope. Therefore, a system of two linear equations representing distinct lines with different slopes has one solution.

**Group C Group F**

•

**Both groups are talking about the pace or the slope. Are they both saying the same thing? (Hook)**

• Can Group C talk about how you determined Serena and Trevon are not moving at the • • • same pace?

Who understood what she said about Trevon’s and Serena’s paces? (Community) Can you say back what he said about the rate of change? (Community) If the pace or slope is different for each person, what does that mean about whether or • not Trevon and Serena are ever at the same depth? (Rigor) What will be true about the slope and the equations if Trevon’s and Serena’s data are the same line (second piece of work)? (Rigor) 40 © 2013 UNIVERSITY OF PITTSBURGH

**Revisiting Your ***Accountable Talk *Prompts with an Eye Toward Problematizing

*Accountable Talk*Prompts with an Eye Toward Problematizing

Revisit your

*Accountable Talk *

prompts.

Have you problematized the mathematics so as to draw students’ attention to the mathematical goal of the lesson? • If you have already problematized the work, then underline the prompt in red. • If you have not problematized the lesson , do so now. Write your problematizing prompt in red at the bottom and indicate where you would insert it in the set of prompts.

*We will be doing a Gallery Walk after we role-play. *

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**Role-Play Our ***Accountable Talk *Discussion

*Accountable Talk*Discussion

• You will have 15 minutes to role-play the discussion of one essential understanding.

• Identify one observer in the group. The observer will keep track of the discussion moves used in the lesson. • The teacher will engage you in a discussion.

*(Note: You are well-behaved students.) *

The goals for the lesson are: to engage all students in the group in developing an understanding of the EU; and to gather evidence of student understanding based on what the student shares during the discussion.

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**Reflecting on the Role-Play: The ***Accountable Talk *Discussion

*Accountable Talk*Discussion

• The observer has two minutes to share observations related to the lessons. The observations should be shared as “noticings.” • Others in the group have one minute to share their “noticings.” 43 © 2013 UNIVERSITY OF PITTSBURGH

**Reflecting on the Role-Play: The ***Accountable Talk *Discussion

*Accountable Talk*Discussion

**(Whole Group Discussion)**

Now that you have engaged in role-playing, what are you now thinking about regarding

*Accountable Talk *

discussions? © 2013 UNIVERSITY OF PITTSBURGH 44

**Zooming In on Problematizing**

**(Whole Group Discussion)**

Do a Gallery Walk. Read each others’ problematizing “hook.” What do you notice about the use of hooks? What role do “hooks” play in the lesson? 45 © 2013 UNIVERSITY OF PITTSBURGH

**Step Back and Application to Our Work**

What have you learned today that you will apply when planning or teaching in your classroom?

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**Summary of Our Planning Process**

Participants: • Identify goals for instruction; – Align Standards for Mathematical Content and Standards for Mathematical Practice with a task.

– Select essential understandings that relate to the Standards for Mathematical Content and Standards for Mathematical Practice.

• Prepare for the Share, Discuss, and Analyze Phase of the lesson.

– Analyze and select student work that can be used to discuss essential understandings of mathematics. – Learn methods of problematizing the mathematics in the lesson. 47 © 2013 UNIVERSITY OF PITTSBURGH